Download Grade 8 Math, Unit 3 Congruence and Similarity 1 Date Created

Middletown Public Schools
Mathematics Unit Planning Organizer
Grade/Course 8
15 instructional days (+5 reteaching/extension)
Duration
Subject
Unit 3
Mathematics
Congruence and Similarity
Big Idea(s)
One can draw reasonable conclusions from repeated observations, and it may be necessary to justify those conclusions using a logical
argument.
Relationships, that undergo transformations, maintain common properties.
How does one make and support a reasonable conclusion regarding a problem?
Essential
Question(s) How do transformations impact mathematical relationships?
Mathematical Practices
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Domain and Standards Overview
Geometry
 Understand congruence and similarity using physical models, transparencies, or geometry software.
Grade 8 Math, Unit 3 Congruence and Similarity
2014
1
Date Created/Revised: November 18,
Priority and Supporting Common Core State Standards
Bold Standards are Priority
8.G.2. Understand that a two-dimensional figure is congruent to
another if the second can be obtained from the first by a sequence of
rotations, reflections and translations; given two congruent figures,
describe a sequence that exhibits the congruence between them.
Explanations and Examples
8.G.2. Examples:
• Is Figure A congruent to Figure A’? Explain how you know.
• Describe the sequence of transformations that results in the
transformation of Figure A to Figure A’.
Grade 8 Math, Unit 3 Congruence and Similarity
2014
2
Date Created/Revised: November 18,
8.G.4. Examples:
8.G.4. Understand that a two-dimensional figure is similar to
another if the second can be obtained from the first by a sequence of
rotations, reflections, translations, and dilations; given two similar
two-dimensional figures, describe a sequence that exhibits the
similarity between them.
• Is Figure A similar to Figure A’? Explain how you know.
• Describe the sequence of transformations that results in the
transformation of Figure A to Figure A’.
Grade 8 Math, Unit 3 Congruence and Similarity
2014
3
Date Created/Revised: November 18,
8.G.1 Verify experimentally the properties of rotations, reflections, and
translations:
8.G.1 Students need multiple opportunities to explore the transformation
of figures so that they can appreciate that points stay the same distance
apart and lines stay at the same angle after they have been rotated,
reflected, and/or translated.
a. Lines are taken to lines, and line segments to line segments of the
same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Students are not expected to work formally with properties of dilations
until high school.
8.G.3.
8.G.3. Describe the effect of dilations, translations, rotations, and
reflections on two-dimensional figures using coordinates.
Dilation: A dilation is a transformation that moves each point along a ray
emanating from a fixed center, and multiplies distances from the center
by a common scale factor. In dilated figures, the dilated figure is similar
to its pre-image.
Translation: A translation is a transformation of an object that moves the
object so that every point of the object moves in the same direction as
well as the same distance. In a translation, the translated object is
congruent to its pre-image. ΔABC has been translated 7 units to the right
and 3 units up. To get from A (1,5) to A’ (8,8), move A 7 units to the
Grade 8 Math, Unit 3 Congruence and Similarity
2014
4
Date Created/Revised: November 18,
right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B +
C also move in the same direction (7 units to the right and 3 units up).
Reflection: A reflection is a transformation that flips an object across a
line of reflection (in a coordinate grid the line of reflection may be the x
or y axis). In a rotation, the rotated object is congruent to its pre-image.
When an object is reflected across the y axis, the reflected x coordinate is
the opposite of the pre-image x coordinate.
Grade 8 Math, Unit 3 Congruence and Similarity
2014
5
Date Created/Revised: November 18,
Rotation: A rotated figure is a figure that has been turned about a fixed
point. This is called the center of rotation. A figure can be rotated up to
360˚. Rotated figures are congruent to their pre-image figures.
Consider when ΔDEF is rotated 180˚ clockwise about the origin. The
coordinates of ΔDEF are D(2,5), E(2,1), and F(8,1). When rotated 180˚,
ΔD’E’F’ has new coordinates D’(-2,-5), E’(-2,-1) and F’(-8,-1). Each
coordinate is the opposite of its pre-image.
8.G.5. Use informal arguments to establish facts about the angle sum and
exterior angle of triangles, about the angles created when parallel lines
are cut by a transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same triangle so that
the sum of the three angles appears to form a line, and give an argument
Grade 8 Math, Unit 3 Congruence and Similarity
2014
6
8.G.5. Examples: Students can informally prove relationships with
transversals.
Show that m∡3+ m∡4+ m∡5= 180˚ if land m are parallel lines and
Date Created/Revised: November 18,
in terms of transversals why this is so.
t1 & t2 are transversals.
∡1+ ∡2+ ∡3= 180˚. Angle and Angle 5 are congruent because they
are corresponding angles (∡5≅∡1). ∡1can be substituted for ∡5.
∡4≅∡2 : because alternate interior angles are congruent. ∡4can be
substituted for ∡2
Therefore m∡3+ m∡4+ m∡5= 180˚
Students can informally conclude that the sum of a triangle is 180º (the
angle-sum theorem) by applying their understanding of lines and
alternate interior angles. In the figure below, line x is parallel to line yz:
Grade 8 Math, Unit 3 Congruence and Similarity
2014
7
Date Created/Revised: November 18,
Angle a is 35º because it alternates with the angle inside the triangle that
measures 35º. Angle c is 80º because it alternates with the angle inside
the triangle that measures 80º. Because lines have a measure of 180º, and
angles a + b + c form a straight line, then angle b must be 65 º (180 – 35
+ 80 = 65). Therefore, the sum of the angles of the triangle are 35º + 65 º
+ 80 º









Concepts
What Students Need to Know
Rotation
Reflection
Translation
Dilation
Congruence
Similarity
Informal proof
Angle sum and exterior angle of triangles
Parallel lines cut by a transversal
o Angles formed
Grade 8 Math, Unit 3 Congruence and Similarity
2014
Skills
What Students Need to Be Able to Do
 VERIFY (Experimentally properties of)
o Rotations
o Reflections
o Translations
o Dilations
 UNDERSTAND (Congruence)
o DESCRIBE (Sequence of rotations,
reflections, translations)
8
Bloom’s Taxonomy Levels
Depth of Knowledge Levels
2
2,3
Date Created/Revised: November 18,

2,3
Angle-angle criterion for similar triangles
 UNDERSTAND (Similarity)
o DESCRIBE (Sequence of rotations,
reflections, translations, dilations)
 DESCRIBE (effect of dilations, translations,
rotations and reflections using
coordinates)
 PROVE (informally)
o angle relationships in parallel lines
cut by a transversal
o sum of angles in a triangle = 180°
2
5
Learning Progressions
The standards below represent prior knowledge and enrichment opportunities for standards in this unit.
Standard
Prerequisite Skills
Accelerate Learning
8.G.2. Understand that a two-dimensional
Experiment with transformations in the plane.
figure is congruent to another if the second
Draw, construct, and describe geometrical
G.CO.1-5
can be obtained from the first by a sequence
figures and describe the relationships between
of rotations, reflections and translations;
Understand congruence in terms of rigid
them. 7.G.1-3
given two congruent figures, describe a
motions. G.CO.6-8
sequence that exhibits the congruence
Use coordinates to prove simple geometric
between them.
Grade 8 Math, Unit 3 Congruence and Similarity
2014
9
Date Created/Revised: November 18,
theorems algebraically. G.GPE.4-7
8.G.4. Understand that a two-dimensional
figure is similar to another if the second can
Draw, construct, and describe geometrical
be obtained from the first by a sequence of
figures and describe the relationships between
rotations, reflections, translations, and
dilations; given two similar two-dimensional them. 7.G.1-3
figures, describe a sequence that exhibits the
similarity between them.
Performance Task
Common Formative Assessment
Grade 8 Math, Unit 3 Congruence and Similarity
2014
Experiment with transformations in the plane.
G.CO.1-5
Understand similarity in terms of similarity
transformations. G.SRT.1-3
Use coordinates to prove simple geometric
theorems algebraically. G.GPE.4-7
Unit Assessments
“Aaron’s Design” http://map.mathshell.org/materials/tasks.php?taskid=361&subpage=apprentice
8.G.2matical ormations impact relationships?ips ain testing)ad to indirect solutions to measurement
problems.10101010101010101010101010101010101010
10
Date Created/Revised: November 18,